{"paper":{"title":"Asymptotically Optimal Tests for One- and Two-Sample Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Threshold tests of relative entropy between empirical distributions are asymptotically optimal for one- and two-sample hypothesis testing.","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Arick Grootveld, Biao Chen, Venkata Gandikota","submitted_at":"2026-01-16T19:20:35Z","abstract_excerpt":"In this work, we revisit the one- and two-sample testing problems: binary hypothesis testing in which one or both distributions are unknown. For the one-sample test, we provide a more streamlined proof of the asymptotic optimality of Hoeffding's likelihood ratio test, which is equivalent to the threshold test of the relative entropy between the empirical distribution and the nominal distribution. The new proof offers an intuitive interpretation and naturally extends to the two-sample test where we show that a similar form of Hoeffding's test, namely a threshold test of the relative entropy bet"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"a similar form of Hoeffding's test, namely a threshold test of the relative entropy between the two empirical distributions is also asymptotically optimal. A strong converse for the two-sample test is also obtained.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes i.i.d. sampling and works in the asymptotic regime where the number of samples tends to infinity; the proofs rely on standard large-deviation properties of empirical distributions that are not re-derived in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Hoeffding's relative entropy threshold test between empirical distributions is asymptotically optimal for both one- and two-sample hypothesis testing, with a strong converse for the two-sample case.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Threshold tests of relative entropy between empirical distributions are asymptotically optimal for one- and two-sample hypothesis testing.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"24fd1adae9b530711d861273c60790943ba4ed0b0b6278f43770bad12f8fe704"},"source":{"id":"2601.11727","kind":"arxiv","version":3},"verdict":{"id":"1386d68b-ec42-480f-80ab-0a40b51e4d95","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T13:03:16.564821Z","strongest_claim":"a similar form of Hoeffding's test, namely a threshold test of the relative entropy between the two empirical distributions is also asymptotically optimal. A strong converse for the two-sample test is also obtained.","one_line_summary":"Hoeffding's relative entropy threshold test between empirical distributions is asymptotically optimal for both one- and two-sample hypothesis testing, with a strong converse for the two-sample case.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes i.i.d. sampling and works in the asymptotic regime where the number of samples tends to infinity; the proofs rely on standard large-deviation properties of empirical distributions that are not re-derived in the abstract.","pith_extraction_headline":"Threshold tests of relative entropy between empirical distributions are asymptotically optimal for one- and two-sample hypothesis testing."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.11727/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}